Coarse-graining Landau–Lifshitz damping

High speed switching in magnetic materials is usually studied with the Landau–Lifshitz (LL) equation, which describes damping through a phenomenological coefficient. The results of micromagnetic calculations based on the LL equation have been observed to depend strongly on the cell size. We take a coarse-graining or renormalization-group approach to this cell size dependence: from a simulation using cell size $\mathrm{L,}$ we look at the dynamics of a cell of size $\text{2L}$ and determine an effective damping coefficient that describes the larger-scale dynamics. This can be thought of as a Green–Kubo calculation of the effective damping coefficient. In principle, this makes it possible to coarse grain from the atomic scale to determine the micromagnetic damping coefficient.